Occupation number.


Subspace Matching
Given the guess orbitals \(\lbrace \psi _{p}^{\mathrm {Y}(0)}\rbrace _{p=1}^{N_{\mathrm {orb}}}\)  (\(\mathrm {Y}=\mathrm {LMO},\mathrm {CMO}\) ), we are ready to perform the CASSCF calculation.
Noticing that each iteration \(k\)  will generate delocalized CMOs \(\lbrace \psi _{i}^{\mathrm {CMO}(k)}\rbrace _{i=1}^{N_{\mathrm {orb}}}\)  and
a proper reassignment of the core, active and virtual spaces has to be performed, we first calculate the occupation numbers of all \(\lbrace \psi _{i}^{\mathrm {CMO}(k)}\rbrace _{i=1}^{N_{\mathrm {orb}}}\) 
in the core space \(\lbrace \psi _p^{\mathrm {Y}(k-1)}\rbrace _{i=1}^{N_\mathrm {c}}\)  of the previous iteration
(cf. Eq. (REF )), so as to identify the core orbitals \(\lbrace \psi _i^{\mathrm {CMO}(k)}\rbrace _{p=1}^{N_\mathrm {c}}\) 
of the present iteration with the largest \(N_{\mathrm {c}}\)  occupation numbers \(\lbrace n_{\mathrm {c},i}\rbrace _{i=1}^{N_{\mathrm {c}}}\) .
Then, the occupation numbers of
the remaining \(\lbrace \psi _{i}^{\mathrm {CMO}(k)}\rbrace _{i=N_\mathrm {c}+1}^{N_{\mathrm {orb}}}\)  in the active space \(\lbrace \psi _p^{\mathrm {Y}(k-1)}\rbrace _{p=N_{\mathrm {c}}+1}^{N_{\mathrm {c}}+K}\) 
are calculated to identify the \(K\)  active orbitals, again with the largest occupation numbers \(\lbrace n_{\mathrm {a},i}\rbrace _{i=1}^{K}\) . The left are obviously virtual orbitals.
In case that some \(\lbrace n_{\mathrm {a},j}\rbrace \)  are very close to the smallest of \(\lbrace n_{\mathrm {a},i}\rbrace _{i=1}^{K}\) , it would indicate the prechosen AOs/pFLMOs are not enough
and should hence be expanded. As a matter of fact, even in this case, the calculation need not be terminated but
can still proceed ahead by just putting such orbitals into the active space,
which will be stabilized in subsequent iterations.
This way, the orbitals of two adjacent iterations
are best matched. To localize the CASSCF-CMOs, they are first expanded as
\((\psi _{\mathrm {c}}^{\mathrm {CMO}(k)},\psi _{\mathrm {a}}^{\mathrm {CMO}(k)},\psi _{\mathrm {v}}^{\mathrm {CMO}(k)})=(\psi _{\mathrm {c}}^{\mathrm {Y}(k-1)},\psi _{\mathrm {a}}^{\mathrm {Y}(k-1)},\psi _{\mathrm {v}}^{\mathrm {Y}(k-1)})\begin{pmatrix}\mathbf {C}_{\mathrm {cc}} & \mathbf {C}_{\mathrm {ca}}&\mathbf {C}_{\mathrm {cv}} \\\mathbf {C}_{\mathrm {ac}} & \mathbf {C}_{\mathrm {aa}}&\mathbf {C}_{\mathrm {av}}\\\mathbf {C}_{\mathrm {vc}} & \mathbf {C}_{\mathrm {va}}&\mathbf {C}_{\mathrm {vv}}\end{pmatrix},\) 
where \(\mathbf {C}_{\mathrm {xy}}=\langle \psi _{\mathrm {x}}^{\mathrm {Y}(k-1)}|\psi _{\mathrm {y}}^{\mathrm {CMO}(k)}\rangle \)  (\(\mathrm {x}, \mathrm {y}=\mathrm {c},\mathrm {a},\mathrm {v}\) ).
The \(\mathbf {T}\)  matrix () can then be constructed so as to transform \(\lbrace \psi _{i}^{\mathrm {CMO}(k)}\rbrace _{i=1}^{N_{\mathrm {orb}}}\)  to \(\lbrace \psi _{i}^{\mathrm {LMO}(k)}\rbrace _{i=1}^{N_{\mathrm {orb}}}\) 
in view of the relation (REF ). Since
the couplings between LMOs are much weaker than those between CMOs, generating CASSCF-LMOs in each iteration can typically facilitate the convergence.
As can be seen from Tables REF  and REF , the converged CASSCF(12,10)-LMOs/CMOs do match the guess LMOs/CMOs for the
case of (C4SH3)-(CH2)10-(C4SH3).
Absolute overlaps between the initial and final CAS(12,10)-LMOs of (C4SH3)-(CH2)10-(C4SH3)

<TABLE>

Absolute overlaps between the initial and final CAS(12,10)-CMOs of (C4SH3)-(CH2)10-(C4SH3)

<TABLE>


Illustrative examples
Except for the multi-state CASPT2[1]} calculations performed with the Molcas program package[2]}, all
calculations were carried out with the BDF program package[3]}, [4]}, [5]}, [6]}, [7]}, [8]}. In particular,
the Xi'an-CI module[8]} in BDF was used to do the multi-state NEVPT2[10]} and SDSPT2[11]}, [12]} calculations.
The valence AOs were generated by SCF calculations of spherical, unpolarized atomic configurations with the same basis functions as used for molecular calculations. Local orientations of the AOs are taken care of automatically.
C-H bond breaking of ethylene
As a first application of the proposed \(\mathbb {i}\) CAS approach,
we examine the breaking of one C-H bond of ethylene, with the lengths of the other three C-H bonds, that of the C-C bond and the \(\angle \) HCH angle set to 1.085 Å, 1.339 Å and 124.8\(^\circ \) , respectively. The basis set is ANO-RCC-VDZP[13]}.
Intuitively, the breaking of a single \(\sigma \) -bond requires only a minimal CAS, i.e.,
two electrons in \(\sigma \)  and \(\sigma ^\ast \) . However, this picture holds only when \(\sigma \)  and \(\sigma ^\ast \)  are the highest occupied (HOMO) and lowest unoccupied (LUMO) molecular orbitals, respectively, during the whole dissociation process.
As can be seen from Fig. REF , the \(\sigma \)  orbital is actually the HOMO-1 (lower than \(\pi \) ) for distances shorter than \(2.0\)  Å and becomes the HOMO only beyond this distance. That is,
there exists a swap of \(\sigma \)  and \(\pi \)  orbitals in the CAS(2,2) across this point. Consequently, the CMO-CAS(2,2) potential energy curve (PEC) is discontinuous nearby \(2.0\)  Å, as can be seen from Fig. REF . A smooth PEC can of course be reproduced by taking all the four frontier orbitals (\(\sigma \) , \(\pi \) , \(\sigma ^\ast \)  and \(\pi ^\ast \) ) as active orbitals, i.e., CAS(4,4).
As pointed out by Malrieu long ago[14]}, the CAS(2,2) calculation can also produce a smooth PEC if localized \(\sigma \)  and \(\sigma ^\ast \)  orbitals are used throughout.
This is indeed confirmed here by the present LMO-\(\mathbb {i}\) CAS(2,2). Interestingly, the present CMO-\(\mathbb {i}\) CAS(2,2), imposed by two pre-CMOs
formed by C\(2\textrm {p}_x\)  and H\(1\textrm {s}\) , can also pick up the two right active orbitals
and hence produces the same PEC as LMO-\(\mathbb {i}\) CAS(2,2). The same findings hold also for the simultaneous breaking of two and even four C-H bonds of ethylene, as
can be seen from Figs. REF  to REF .
Although this example is very simple, it does reflect an important merit of \(\mathbb {i}\) CAS:
the active space spanned by the prechosen AOs hold the same for all geometries.
<FIGURE><FIGURE><FIGURE><FIGURE><FIGURE><FIGURE>
Electronic Structure of CoIII(diiminato)(NPh)
As a more stringent test of \(\mathbb {i}\) CAS, we investigate the spin-state energetics of the low-coordinate imido complex CoIII(diiminato)(NPh),
which is a simplified model of diamagnetic complex CoIII(nacnac)(NAd) [NB: nacnac=an anion of 2,4-bis(2,6-dimethylphenylimido) pentane; Ad=1-adamantyl][15]}.
The density functional theory study [16]} shows that there should be low-lying paramagnetic excited states besides the diamagnetic ground state, which has been calibrated by CAS(10,10)-based CASPT2[17]}.
The major concern here is how to construct an appropriate CAS. On the practical side,
the spin-free exact two-component (sf-X2C) Hamiltonian[18]}, [19]} in conjunction with the ANO-RCC-VDZP basis sets (Co/5s4p2d1f; N,C/3s2p1d; H/2s1p)[13]}
(as used in Ref. CoN-1) is employed here. Twenty-nine core orbitals are frozen in the treatment of dynamic correlation, leaving in total 134 correlated
electrons. In the CASPT2 calculations (with the spin-free Douglas-Kroll-Hess (DKH) Hamiltonian[21]}), an imaginary level shift of 0.1 is used to remove intruder states.
The singlet ground state geometry of CoIII(diiminato)(NPh) was optimized with sf-X2C-PBE0/def2-SVP[22]}, [23]},
without symmetry restrictions. As can be seen from Fig. REF , the Co atom is located almost at the center of the molecular skeleton, the imido nitrogen (N\(_{\textrm {imido}}\) ) is on the y axis,
the 1,3-propanediiminato and phenylimido groups are in the xy plane, whereas the two terminal phenyl groups are perpendicular to the xy plane (see Supporting Information for
the Cartesian coordinates). The crucial coordinate of CoIII(diiminato)(NPh) is the Co-N\(_{\textrm {imido}}\)  bond, which was optimized to be 1.612 Å,
very close to the CASPT2 value of 1.632 Å[17]}.
<FIGURE><FIGURE>To initiate the “top-down, least-change” localization procedure (see Sec. REF ), the molecule is partitioned into four fragments (see Fig. REF ), with the dangling bonds saturated with the modified[25]} PHO (projected hybrid orbital) boundary atoms[26]}. While subsystems 1-3 are obviously closed-shell systems,
subsystem 4 can be in singlet, triplet or quintet. After the HF/ROHF calculations of
these subsystems, the orthonormal pFLMOs are generated[27]}, [28]}, which are taken as the initial guess for the HF/ROHF calculations of the whole systems.
The FLMOs are finally generated according to the “top-down, least-change” scheme. Since such FLMOs
have one-to-one correspondence with the pFLMOs, the active orbitals can be chosen simply from the subsystem pFLMOs. However, to make the \(\mathbb {i}\) CAS calculations
“more difficult”, we still start with valence AOs. Looking at the chemical structure of the system, it should be easy to guess that, apart from
the five 3d orbitals of Co, the five 2p\(_z\)  orbitals of the C and N atoms of the 5-membered ring as well as the 2p\(_x\)  and 2p\(_z\)  of N\(_{\textrm {imido}}\) 
are most relevant for the chemical environment of Co, since the 2p\(_y\)  orbital of N\(_{\textrm {imido}}\)  participates in the bonding with the 6-membered ring.
In addition to these valence AOs, the five 4d orbitals of Co can further be included to account for the so-called “double d-shell effect”
resulting from physical intuitions[29]}.
In essence, putting Co 4d into the active space amounts to shifting the inaccurate, perturbative treatment of their
radial correlation effect to the exact, variational treatment. We then have in total 17 valence AOs (and hence 17 pre-LMOs/CMOs) along with 16 active electrons
(6 for Co\(^{3+}\)  (3d\(^6\) ), 6 for the anionic 5-membered ring, and 4 for \(:\) N\(_{\textrm {imido}}^{2-}\) )
according to minimal chemical and physical intuitions. Taking the 8 doubly occupied and 9 unoccupied pre-LMOs as probes, precisely
17 HF-LMOs (see Fig. REF ) can be selected as initial guess for \(\mathbb {i}\) CAS(16,17).
As can be seen from Tables REF  and REF ,
the large absolute overlaps between the pre-LMOs and HF-LMOs do support the selection. The same findings also hold for the triplet and quintet states (see
Supporting Information). Nevertheless, the CAS(16,17) calculations are too expensive (which, however, can readily be
performed with the iCISCF algorithm[30]} that employs the selected iterative configuration interaction (iCI)[31]}, [32]}, [33]} as the CAS solver).
The first attempt to reduce CAS(16,17) is to exclude the five 2p\(_z\)  orbitals of the 5-membered ring, thereby leading to CAS(10,12).
As a matter of fact, this amounts just to deleting from Fig. REF  as well as Tables REF  and REF 
the \(\pi -1\) , \(\pi -2\) , \(\pi -2\) , \(\pi ^\ast -1\) , and \(\pi ^\ast -2\)  orbitals that are
localized on the 5-membered ring.
It was also attempted[17]} to further remove the 4d\(_{xy}\)  and 4d\(_{yz}\)  orbitals from CAS(10,12), for the 3d\(_{y^2}\) , 3d\(_{x^2-z^2}\)  and 3d\(_{xz}\) 
orbitals are doubly occupied in the singlet ground state, such that only 4d\(_{y^2}\) , 4d\(_{x^2-z^2}\)  and 4d\(_{xz}\)  are relevant for the “double d-shell effect”.
A more dramatic simplification is to exclude all the 4d orbitals from CAS(10,12), thereby leading to CAS(10,7).
These two cases amount to deleting from Fig. REF  the last two and five orbitals involving Co 4d, respectively.
Having prepared the initial guess, two-state averaged \(\mathbb {i}\) CAS (SA2-\(\mathbb {i}\) CAS) calculations are performed for each spin state.
As can be seen from Table REF , the converged \(\mathbb {i}\) CAS(10,12) orbitals are indeed very close to the initial ones. Again, the same
findings also hold for the triplet and quintet states (see Supporting Information). For completeness,
the leading configurations of the CAS wave functions are given in Table REF .
The vertical excitation energies calculated by \(\mathbb {i}\) CAS, CASPT2, NEVPT2, and SDSPT2 are documented in Table REF .
It can first be seen that, at the CASSCF level, CAS(10,7) is far from being sufficient: it even predicts that the two nearly degenerate \(^3A\)  states
are lower than the lowest singlet state by 0.3 eV; CAS(10,10) is also imperfect: although it does predict correctly the ground state[15]}, the excitation energies
of the remaining states are too high, by 0.3 to 0.6 eV. In contrast, CAS(10,12) is essentially saturated, with the predicted excitation energies differing
from the CAS(16,17) ones by at most 0.15 eV. It can also be seen that the imperfections in CAS(10,10) and even CAS(10,7) are essentially removed by the
subsequent CASPT2/NEVPT2/SDSPT2 treatment of dynamic correlation, indicating that the “double d-shell effect” plays a very minor role here. As a matter of fact,
there exists very little dynamic correlation (ca. 0.1 eV) beyond CAS(10,12), as compared with CAS(10,12)-NEVPT2/SDSPT2.
This can be explained by the fact that the spin states investigated here all arise from spin flips within the same Co 3d manifold (cf. Table REF ), for which dynamic correlation
effects are largely cancelled out for the relative energies. The situation may be different for states involving changes in the occupations of \(n\) d (and \(n\) f) orbitals,
where substantial differential correlation effects may be experienced. Note also that there exist substantial differences (ca. 0.4 eV) between the present and the previous[17]}
CAS(10,10)-CASPT2 excitation energies for the two \(^5A\)  states. Although the same CASPT2 code[2]} is used in both cases, the present
calculation takes the SA2-iCAS(10,10) LMOs as input.
To further examine the efficacy of \(\mathbb {i}\) CAS in maintaining the same CAS for all geometries, the rigid stretching of the Co-N\(_{\textrm {imido}}\)  bond (keeping the other atoms fixed) is considered.
The SA2-\(\mathbb {i}\) CAS(10,12) PECs for the six states are plotted in Fig. REF . The smoothness of the PECs is obvious.
While the Co-N\(_{\textrm {imido}}\)  bond in the \(1^1A\)  and \(2^1A\)  states starts to dissociate around 4.5 Å, that in the four triplet and quintet states does not even up to 10 Å.
It is also of interest to see that the four triplet and quintet states become degenerate beyond 2.3 Å. Finally,
the equilibrium Co-N\(_{\textrm {imido}}\)  distances \(R_e\)  are given in Table REF .
Again, there exist significant differences (ca. 0.04 Å) between the present and the previous[17]} CASPT2 values for \(R_e\) , due to the use of different CAS(10,10) orbitals.
Absolute overlaps between the eight doubly occupied pre-LMOs and all doubly occupied molecular HF-LMOs for singlet CoIII(diiminato)(NPh)

<TABLE>
a
\(b_\parallel \)  and \(b_\perp \)  denote Co(3d)-N\(_{\textrm {imido}}\)  in plane \(\pi \)  and out-of-plane \(\pi \) ; \(\pi \) -1, \(\pi \) -2 and \(\pi \) -3 correspond to localized \(\pi \)  of 1,3-propanediiminato.
b
Columns (core orbitals) with values smaller than 0.059.
c
Occupation number.


Absolute overlaps between the nine unoccupied pre-LMOs and all unoccupied molecular HF-LMOs for singlet CoIII(diiminato)(NPh)

<TABLE>
a
\(b_\parallel ^\ast \)  and \(b_\perp ^\ast \)  denote Co(3d)-N\(_{\textrm {imido}}\)  in plane and out-of-plane \(\pi ^\ast \) ; \(\tilde{b}_\parallel ^\ast \)  and \(\tilde{b}_\perp ^\ast \)  denote Co(4d)-N\(_{\textrm {imido}}\)  in plane and out-of-plane \(\pi ^\ast \) ; \(\pi ^\ast \) -1 and \(\pi ^\ast \) -2 correspond to localized \(\pi ^\ast \)  of 1,3-propanediiminato.
b
Columns (virtual orbitals) with values smaller than 0.073.
c
Occupation number.


Absolute overlaps between the initial and final SA2-\(\mathbb {i}\) CAS(10,12) LMOs for singlet CoIII(diiminato)(NPh)

<TABLE>

Leading configurations of HF/ROHF and \(\mathbb {i}\) CAS calculations of CoIII(diiminato)(NPh)

<TABLE>
a
\(b_\parallel \)  (\(b_\parallel ^\ast \) ) and \(b_\perp \)  (\(b_\perp ^\ast \) ) denote Co(4d)-N\(_{\textrm {imido}}\)  in plane and out-of-plane \(\pi \)  (\(\pi ^\ast \) ), respectively;
\(\tilde{b}_\perp ^\ast \)  denotes Co(4d)-N\(_{\textrm {imido}}\)  out-of-plane \(\pi ^\ast \)  with \(\textrm {4d}_{yz}\)  as the major.
b
HF/ROHF total energies relative to the singlet (-2361.437053 a.u.); weights of leading configurations for CASSCF.


Vertical excitation energies (in eV) of CoIII(diiminato)(NPh)
calculated at the singlet ground state geometry (see Supporting Information)

<TABLE>
a
CAS1, CAS2, CAS3 and CAS4 denote CAS(10,7), CAS(10,10), CAS(10,12) and CAS(16,17), respectively.
b
Calculated with the iCISCF approach[30]}.
c
Values in parentheses are results from Ref. CoN-1.


<FIGURE>Equilibrium distances (in Å) for the rigid dissociation of Co-N\(_{\textrm {imido}}\)  in
CoIII(diiminato)(NPh)

<TABLE>

Conclusions and outlook
An \(\mathbb {i}\) CAS method has been proposed to serve as an initial step for describing strongly correlated systems of electrons.
It has several features: (1) the active orbitals can be selected automatically by starting with prechosen valence/core atomic/fragmental orbitals.
The only requirement lies in that such atomic/fragmental orbitals span a space that is sufficient for the target many-electron states,
which hold for most organic molecules or transition metal complexes. (2) The CASSCF orbitals can readily be localized in each iteration so as to
facilitate the matching of core, active and virtual subspaces.
(3) The CAS can be maintained the same for all geometries, which is of vital importance for the scanning of potential energy surfaces.
(4) Further combined with the iterative vector interaction (iVI) approach[40]}, [41]} for directly accessing interior roots of the CI eigenvalue problem,
the CASSCF calculation can be dictated to converge to a predesignated chare/spin configuration composed of some special motifs (e.g., metal
valence d/f orbitals, atomic core orbitals or aromatic rings), a point that is to be investigated in future though.

Acknowledgement
This work was supported by the National Natural Science Foundation of China (Grant Nos. 21833001 and 21973054),
Natural Science Basic Research Plan of Shaanxi Province (Grant No. 2019JM-196),
Mountain Tai Climb Program of Shandong Province, and Key-Area Research and Development Program of Guangdong Province (Grant No. 2020B0101350001).

Data Availability Statement
The Cartesian coordinates of the CoIII(diiminato)(NPh) as well as various plots of the CASSCF orbitals
are documented in the Supporting Information.
